Constraint Systems

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Answer Key & Explanations

Constraint Systems · Grade 9 · Worksheet 2

1. Isabella is buying snacks for a party. She has $72 to spend. Chips cost $7 per bag and drinks cost $2 per bottle. She needs at least 12 items total and at least 3 bags of chips. Write the system of inequalities that represents these constraints, using x for bags of chips and y for bottles of drinks. Answer: 7x + 2y ≤ 72, x + y ≥ 12, x ≥ 3, y ≥ 0 Solution: Write the cost constraint. Chips cost $7 each, drinks cost $2 each, and the total budget is $72. So 7x + 2y ≤ 72.
   Full step-by-step solution

   Step 1: Write the cost constraint. Chips cost $7 each, drinks cost $2 each, and the total budget is $72. So 7x + 2y ≤ 72. Step 2: Write the total items constraint. She needs at least 12 items total, so x + y ≥ 12. Step 3: Write the minimum chips constraint. She needs at least 3 bags of chips, so x ≥ 3. Step 4: Write the non-negativity constraint for drinks. Since she can't buy negative bottles, y ≥ 0. Step 5: Combine all constraints to form the system: 7x + 2y ≤ 72, x + y ≥ 12, x ≥ 3, y ≥ 0.

2. Aroha is planning a community garden with two types of vegetable beds. She can spend at most $150 on materials. Each large bed costs $12 and each small bed costs $9. She needs at least 10 beds total and wants at least 3 large beds. Write the system of inequalities that represents these constraints, where x is the number of large beds and y is the number of small beds. Answer: 12x + 9y ≤ 150, x + y ≥ 10, x ≥ 3, x ≥ 0, y ≥ 0 Solution: Write the cost constraint: Each large bed costs $12 and each small bed costs $9, with a maximum budget of $150. This gives: 12x + 9y ≤ 150 Write the total beds constraint: Aroha needs at least 10 beds total.
   Full step-by-step solution

   Step 1: Write the cost constraint: Each large bed costs $12 and each small bed costs $9, with a maximum budget of $150. This gives: 12x + 9y ≤ 150 Step 2: Write the total beds constraint: Aroha needs at least 10 beds total. This gives: x + y ≥ 10 Step 3: Write the large beds constraint: She wants at least 3 large beds. This gives: x ≥ 3 Step 4: Add non-negativity constraints: The number of beds cannot be negative, so x ≥ 0 and y ≥ 0 Step 5: Combine all constraints to form the system: 12x + 9y ≤ 150 x + y ≥ 10 x ≥ 3 x ≥ 0 y ≥ 0 The system of inequalities is: 12x + 9y ≤ 150, x + y ≥ 10, x ≥ 3, x ≥ 0, y ≥ 0

3. A music festival is planning food truck placement. They have space for at most 15 food trucks total. They need at least 4 dessert trucks to satisfy demand, and the number of savory trucks must be at least double the number of dessert trucks. The festival organizers want to maximize revenue, and they estimate each dessert truck generates $800 per day while each savory truck generates $1200 per day. How many dessert trucks should they allow to maximize daily revenue while satisfying all constraints? Answer: 4 Solution: Let d = number of dessert trucks, s = number of savory trucks - Total trucks: d + s ≤ 15 - Minimum dessert trucks: d ≥ 4 - Savory to dessert ratio: s ≥ 2d - Non-negative: d ≥ 0, s ≥ 0 Revenue function: R = 800d + 1200s - Point A: d = 4, s = 8 (minimum dessert, minimum savory) - Point B: d = 4, s…
   Full step-by-step solution

   Step 1: Let d = number of dessert trucks, s = number of savory trucks Step 2: Write the constraints as inequalities: - Total trucks: d + s ≤ 15 - Minimum dessert trucks: d ≥ 4 - Savory to dessert ratio: s ≥ 2d - Non-negative: d ≥ 0, s ≥ 0 Step 3: Revenue function: R = 800d + 1200s Step 4: Find corner points of the feasible region: - Point A: d = 4, s = 8 (minimum dessert, minimum savory) - Point B: d = 4, s = 11 (minimum dessert, maximum total: 4 + 11 = 15) - Point C: d = 5, s = 10 (s = 2d constraint, total 15) Step 5: Calculate revenue at each corner point: - Point A: R = 800(4) + 1200(8) = 3200 + 9600 = 12800 - Point B: R = 800(4) + 1200(11) = 3200 + 13200 = 16400 - Point C: R = 800(5) + 1200(10) = 4000 + 12000 = 16000 Step 6: The maximum revenue is $16,400 at point B with 4 dessert trucks and 11 savory trucks. The answer is 4.

4. A right triangle is drawn on a coordinate plane with vertices at A(0,0), B(8,0), and C(0,6). A circle is circumscribed around this triangle such that all three vertices lie on the circle's circumference. What is the area of this circumscribed circle? (Use π = 3.14) Answer: 78.5 Solution: Identify that for a right triangle, the hypotenuse is the diameter of the circumscribed circle. The hypotenuse is between points B(8,0) and C(0,6).
   Full step-by-step solution

   Step 1: Identify that for a right triangle, the hypotenuse is the diameter of the circumscribed circle. Step 2: The hypotenuse is between points B(8,0) and C(0,6). Step 3: Calculate the length of the hypotenuse using the distance formula: sqrt((8-0)^2 + (0-6)^2) = sqrt(64 + 36) = sqrt(100) = 10. Step 4: This length (10) is the diameter of the circle. Step 5: The radius is half the diameter: 10/2 = 5. Step 6: Calculate the area of the circle using the formula: Area = π × radius^2 = 3.14 × 5^2 = 3.14 × 25 = 78.5. The answer is 78.5.

5. Aisha is designing a rectangular mural for her school's art exhibition. The mural must have a perimeter of exactly 36 meters. She wants the length to be at least 4 meters more than the width, and the area must be greater than 80 square meters to accommodate her design. Write a system of inequalities that represents these constraints, using L for length and W for width. Answer: 2L + 2W = 36, L ≥ W + 4, L × W > 80 Solution: The perimeter is 36 meters, so 2L + 2W = 36 The length is at least 4 meters more than the width, so L ≥ W + 4 The area must be greater than 80 square meters, so L × W > 80 The complete system is: 2L + 2W = 36, L ≥ W + 4, L × W > 80
   Full step-by-step solution

   Step 1: The perimeter is 36 meters, so 2L + 2W = 36 Step 2: The length is at least 4 meters more than the width, so L ≥ W + 4 Step 3: The area must be greater than 80 square meters, so L × W > 80 Step 4: The complete system is: 2L + 2W = 36, L ≥ W + 4, L × W > 80

6. Isabella is organizing a school fundraiser and has a budget of $72. She needs to buy at least 12 items total, including notebooks and pens. Each notebook costs $7 and each pen costs $2. Let x represent notebooks and y represent pens. Write the system of inequalities that represents these constraints. Answer: 7x + 2y ≤ 72, x + y ≥ 12, x ≥ 0, y ≥ 0 Solution: Identify the cost constraint: Each notebook costs $7 and each pen costs $2, with a total budget of $72. This gives us: 7x + 2y ≤ 72 Identify the quantity constraint: At least 12 items total are needed.
   Full step-by-step solution

   Step 1: Identify the cost constraint: Each notebook costs $7 and each pen costs $2, with a total budget of $72. This gives us: 7x + 2y ≤ 72 Step 2: Identify the quantity constraint: At least 12 items total are needed. This gives us: x + y ≥ 12 Step 3: Identify the non-negativity constraints: Since we cannot have negative notebooks or pens, we have: x ≥ 0 and y ≥ 0 Step 4: Combine all constraints to form the system: 7x + 2y ≤ 72 x + y ≥ 12 x ≥ 0 y ≥ 0 The complete system of inequalities is: 7x + 2y ≤ 72, x + y ≥ 12, x ≥ 0, y ≥ 0

7. √(x² - 4x + 4) = 5, solve for x Answer: 7 Solution: Recognize that x² - 4x + 4 is a perfect square trinomial Factor the expression: x² - 4x + 4 = (x - 2)² The equation becomes √((x - 2)²) = 5 The square root of a square gives the absolute value: |x - 2| = 5 Solve the absolute value equation: x - 2 = 5 or x - 2 = -5 For x - 2 = 5: x = 7 For x - 2…
   Full step-by-step solution

   Step 1: Recognize that x² - 4x + 4 is a perfect square trinomial Step 2: Factor the expression: x² - 4x + 4 = (x - 2)² Step 3: The equation becomes √((x - 2)²) = 5 Step 4: The square root of a square gives the absolute value: |x - 2| = 5 Step 5: Solve the absolute value equation: x - 2 = 5 or x - 2 = -5 Step 6: For x - 2 = 5: x = 7 Step 7: For x - 2 = -5: x = -3 Step 8: Check both solutions in the original equation Step 9: For x = 7: √(49 - 28 + 4) = √25 = 5 ✓ Step 10: For x = -3: √(9 + 12 + 4) = √25 = 5 ✓ Both solutions are valid, but the problem asks for one solution, so we can use x = 7.